#include "jumps.h"
#include <bits/stdc++.h>
using namespace std;
vector<int> next_greater(const vector<int> &v) {
int n = v.size();
vector<int> ans(n, -1);
vector<int> possible_next{n - 1};
for (int i = n - 2; i >= 0; --i) {
while (!possible_next.empty() && v[i] >= v[possible_next.back()])
possible_next.pop_back();
if (!possible_next.empty())
ans[i] = possible_next.back();
possible_next.push_back(i);
}
return ans;
}
using interval = pair<int, int>;
interval onion(interval a, interval b) {
return {min(a.first, b.first), max(a.second, b.second)};
}
interval intersection(interval a, interval b) {
return {max(a.first, b.first), min(a.second, b.second)};
}
bool has_overlap(interval a, interval b) {
return !(a.second < b.first || b.second < a.first);
}
// fn ((a, b), (a, b)) { (a, b) }
// let onion((x, y), (z, w)) = (min(x, z), max(y, w))
struct Jumper {
vector<vector<int>> tree_ch;
vector<vector<int>> locations;
vector<int> depth;
vector<pair<int, int>> reachable_from;
vector<vector<pair<int, int>>> ranges_crossed;
void init_reachable_from(int v) {
reachable_from[v] = {v, v};
for (int ch : tree_ch[v]) {
depth[ch] = depth[v] + 1;
init_reachable_from(ch);
reachable_from[v] = onion(reachable_from[v], reachable_from[ch]);
}
}
Jumper() {}
Jumper(int root, const vector<int> &tree,
const vector<pair<int, int>> init_ranges)
: tree_ch(tree.size()), locations(20, vector(tree.size(), -1)),
depth(tree.size()),
reachable_from(tree.size(), {-1, -1}),
ranges_crossed(
20, vector(tree.size(), pair<int, int>{tree.size() + 100, -1})) {
for (int i = 0; i < tree.size(); ++i)
if (tree[i] != -1) {
tree_ch[tree[i]].push_back(i);
locations[0][i] = tree[i];
}
init_reachable_from(root);
ranges_crossed[0] = init_ranges;
for (int i = 0; i < tree.size(); ++i) {
int layer = 0;
// cerr << "layer " << layer << " " << i << " -> " << locations[layer][i]
// << " (" << ranges_crossed[layer][i].first << ", "
// << ranges_crossed[layer][i].second << ")" << endl;
}
for (int layer = 1; layer < locations.size(); ++layer) {
for (int i = 0; i < tree.size(); ++i)
if (locations[layer - 1][i] != -1) {
locations[layer][i] = locations[layer - 1][locations[layer - 1][i]];
ranges_crossed[layer][i] =
onion(ranges_crossed[layer - 1][i],
ranges_crossed[layer - 1][locations[layer - 1][i]]);
// cerr << "layer " << layer << " " << i << " -> " << locations[layer][i]
// << " (" << ranges_crossed[layer][i].first << ", "
// << ranges_crossed[layer][i].second << ")" << endl;
}
}
}
int last_inside(int v, interval i) {
int cv = v;
for (int layer = locations.size() - 1; layer >= 0; --layer) {
if (locations[layer][cv] != -1 && i.first <= locations[layer][cv] &&
locations[layer][cv] <= i.second)
cv = locations[layer][cv];
}
return cv;
}
// {vertex, # of steps taken, interval visited}
tuple<int, int, interval> last_before_overlapping(int v, interval i) {
int cv = v;
auto crange = ranges_crossed[0][v];
int dist = 0;
for (int layer = locations.size() - 1; layer >= 0; --layer) {
if (locations[layer][cv] != -1 &&
!has_overlap(onion(crange, ranges_crossed[layer][cv]), i)) {
crange = onion(crange, ranges_crossed[layer][cv]);
cv = locations[layer][cv];
dist += (1 << layer);
}
}
return {cv, dist, crange};
}
int direct_len(int v, interval i) {
auto [dst, dist, _] = last_before_overlapping(v, i);
if (i.first <= locations[0][dst] && locations[0][dst] <= i.second) {
return dist + 1;
} else
return -1;
}
int parent(int v) { return locations[0][v]; }
interval reachable_range(int v) { return reachable_from[v]; }
int lca(int i, int j) {
if (i == j) return i;
if (depth[i] < depth[j]) swap(i, j);
for (int layer = locations.size() - 1; layer >= 0; --layer) {
if (locations[layer][i] != -1 && depth[locations[layer][i]] >= depth[j]) i = locations[layer][i];
}
if (i == j) return i;
for (int layer = locations.size() - 1; layer >= 0; --layer) {
if (locations[layer][i] != -1 && locations[layer][i] != locations[layer][j]) {
i = locations[layer][i];
j = locations[layer][j];
}
}
return locations[0][i];
}
};
Jumper fast, slow;
vector<int> H;
void init(int N, std::vector<int> _H) {
H = _H;
// impl detail so ng and pg are always defined on nodes that matter
vector<int> ng = next_greater(H);
reverse(H.begin(), H.end());
vector<int> pg = next_greater(H);
reverse(H.begin(), H.end());
reverse(pg.begin(), pg.end());
for (int &v : pg)
v = (v == -1 ? -1 : N - 1 - v);
vector<int> fast_tree(N, -1), slow_tree(N, -1);
for (int i = 0; i < N; ++i) {
if (ng[i] == -1 && pg[i] == -1)
continue;
else if (ng[i] == -1 || pg[i] == -1) {
int par = max(ng[i], pg[i]);
fast_tree[i] = slow_tree[i] = par;
} else {
pair<int, int> target1 = {H[ng[i]], ng[i]}, target2 = {H[pg[i]], pg[i]};
if (target1 > target2)
swap(target1, target2);
fast_tree[i] = target2.second;
slow_tree[i] = target1.second;
}
// cerr << "jump targets: " << fast_tree[i] << " " << slow_tree[i] << endl;
}
int max_idx = distance(begin(H), max_element(begin(H), end(H)));
// get ranges for the slow tree
vector slow_tree_init_ranges(N, pair{-1, -1});
for (int i = 0; i < N; ++i) {
slow_tree_init_ranges[i] = {min(i, slow_tree[i]), max(i, slow_tree[i])};
}
// cerr << " == processing slow tree " << endl;
slow = Jumper(max_idx, slow_tree, slow_tree_init_ranges);
// from traversal info on the slow tree, get ranges for the fast tree
vector fast_tree_init_ranges(N, pair{-1, -1});
for (int i = 0; i < N; ++i) {
if (fast_tree[i] != -1) {
pair ft_int{fast_tree[i], fast_tree[i]};
pair slow_tree_int = get<2>(slow.last_before_overlapping(i, ft_int));
fast_tree_init_ranges[i] = onion(slow_tree_int, ft_int);
}
}
// cerr << " == processing fast tree " << endl;
fast = Jumper(max_idx, fast_tree, fast_tree_init_ranges);
}
int brute_force_minimum_jumps(int A, int B, int C, int D) {
vector<int> previ(H.size(), -1), nexti(H.size(), -1);
for (int cur = 0; cur < H.size(); ++cur) {
for (int i = cur - 1; i >= 0; --i)
if (H[i] > H[cur]) {
previ[cur] = i;
break;
}
for (int j = cur + 1; j < H.size(); ++j)
if (H[j] > H[cur]) {
nexti[cur] = j;
break;
}
}
int ans = -1;
for (int start = A; start <= B; ++start) {
vector<int> dist(H.size(), -1);
queue<int> q;
q.push(start);
dist[start] = 0;
while (!q.empty()) {
int cur = q.front();
q.pop();
if (C <= cur && cur <= D) {
ans = (ans == -1 ? dist[cur] : min(ans, dist[cur]));
}
int prev = previ[cur], next = nexti[cur];
if (prev != -1 && dist[prev] == -1) {
dist[prev] = dist[cur] + 1;
q.push(prev);
}
if (next != -1 && dist[next] == -1) {
dist[next] = dist[cur] + 1;
q.push(next);
}
}
}
return ans;
}
int minimum_jumps_single_source(int src, int C, int D, interval possible_sources) {
// cerr << "MJSS " << src << endl;
if (possible_sources.first > src || possible_sources.second < src) return -1;
// cerr << " starting from " << src << endl;
int ans = -1;;
auto consider = [&] (int v) {
if (ans == -1 || (v != -1 && v < ans)) ans = v;
};
// take as many fast edges as possible before our path, traced along
// the slow edges, hits {C, D}
auto [v1, dist1, _] = fast.last_before_overlapping(src, {C, D});
// cerr << " phase 1: " << v1 << " in " << dist1 << " jumps" << endl;
int v1p = fast.parent(v1);
if (C <= v1p && v1p <= D) {
// case 1: another fast edge gets us there
consider(dist1 + 1);
} else {
// case 2: we get there using only slow edges
auto [v2, dist2, __] = slow.last_before_overlapping(v1, {C, D});
int pot_dst = slow.parent(v2);
// cerr << " phase 2: " << v2 << " in " << dist2 << " jumps -> " << pot_dst << endl;
if (C <= pot_dst && pot_dst <= D)
consider(dist1 + dist2 + 1);
// case 3: we need to take another fast edge
// MAGIC CLAIM: This only happens O(sqrt(n)) times
int further_dist = minimum_jumps_single_source(v1p, C, D, possible_sources);
if (further_dist != -1) consider(minimum_jumps_single_source(v1p, C, D, possible_sources) + dist1 + 1);
}
// cerr << "MJSS " << src << " ret " << ans << endl;
return ans;
}
int minimum_jumps_real(int A, int B, int C, int D) {
interval reachable_from_dst =
slow.reachable_range(slow.lca(C, D));
// cerr << "rfdst " << reachable_from_dst.first << " " << reachable_from_dst.second << endl;
auto [realA, realB] = intersection({A, B}, reachable_from_dst);
if (realA > realB) return -1;
int src = slow.lca(realA, realB);
return minimum_jumps_single_source(src, C, D, reachable_from_dst);
}
int minimum_jumps(int A, int B, int C, int D) {
int ans = minimum_jumps_real(A, B, C, D);
/* int bans = brute_force_minimum_jumps(A, B, C, D);
cerr << "expected " << bans << " got " << ans << endl;
assert(ans == bans);*/
return ans;
}
